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I was watching a video on the derivation of the Maxwell-Boltzmann distribution function which would eventually lead to:

$$\frac{e^{-\beta \cdot \epsilon_i}}{\sum_{i=0}^n e^{-\beta \cdot \epsilon_i}}$$

To do this, initially, the number of possible permutations ##\Omega## of a total of ##N## molecules in an isolated system distributed over ##n## energy compartments is first concluded:

$$\Omega = \frac{N!}{n_1! n_2! n_3! … n_n!}$$

It is said that one would have to solve this equation for the maximum amount of possible permutations because a state of thermal equilibrium should have the highest probability.

I have 3 questions regarding this permutations formula:

**1.**I still have difficulty grasping the concept why a system in thermal equilibrium must have the maximum amount of possible permutations. Why isn’t it possible for a system in thermal equilibrium to have less?

**2.**To get the maxima, the formula is written in terms of ##ln(..)## and differentiated so that one can solve for 0. However, I can already see from the formula that the maximum amount of possible permutations is when each compartment ##n_i## contains just 1 molecule, so that there are ##N## compartments just as there are ##N## molecules of the whole system. Why can't it be reasoned this way?

**3.**I understand that this derivation is classical and thus ignoring the discrete quantum energy levels between the energy compartments. However, shouldn’t this mean that the difference in energies between the energy compartments is continuous so that there are an infinite amount of energy compartments? If so, how is it then possible to calculate the amount of permutations with a formula that shows a limited number of energy compartments?

Really looking forward to some clarifications on these questions.